Over medium heat, sauté garlic and ginger in olive oil for 30 seconds until fragrant. Add ground chicken, soy sauce, rice vinegar, chili garlic sauce, sesame oil, salt and pepper, and sauté just until chicken is cooked through (about 4-5 min).
Turn heat down to low, and add mushrooms, sesame seeds and cabbage, cooking for another 5 min. Remove from heat and stir in green onions.
Unwrap wonton wrappers from packaging and set out on a large, very lightly floured surface. spoon about 1 tbsp of filling into each wrapper, and use water to seal the edges, pressing with a fork if needed. Continue until you've used all the filling or until you have enough potstickers.
Over med-high heat, heat olive oil in a large pan. Cook potstickers one layer at a time for about 45 seconds each side (more if desired - just be sure to watch they don't burn!)
Serve right away, or freeze any cooked portions for a future date - they can be reheated in a 400 F oven for 15-20 minutes. The filling also freezes well if you want to make more with fresh wonton wrappers at a later date!